Multirate Method Research Articles

Explicit stabilized multirate methods for the monodomain model in cardiac electrophysiology

Fully explicit stabilized multirate (mRKC) methods are well-suited for the numerical solution of large multiscale systems of stiff ordinary differential equations thanks to their improved stability properties. To demonstrate their efficiency for the numerical solution of stiff, multiscale, nonlinear parabolic PDE's, we apply mRKC methods to the monodomain equation from cardiac electrophysiology. In doing so, we propose an improved version, specifically tailored to the monodomain model, which leads to the explicit exponential multirate stabilized (emRKC) method. Several numerical experiments are conducted to evaluate the efficiency of both mRKC and emRKC, while taking into account different finite element meshes (structured and unstructured) and realistic ionic models. The new emRKC method typically outperforms a standard implicit-explicit baseline method for cardiac electrophysiology. Code profiling and strong scalability results further demonstrate that emRKC is faster and inherently parallel without sacrificing accuracy.

Performance of explicit and IMEX MRI multirate methods on complex reactive flow problems within modern parallel adaptive structured grid frameworks

Large-scale multiphysics simulations are computationally challenging due to the coupling of multiple processes with widely disparate time scales. The advent of exascale computing systems exacerbates these challenges since these systems enable ever-increasing size and complexity. In recent years, there has been renewed interest in developing multirate methods as a means to handle the large range of time scales, as these methods may afford greater accuracy and efficiency than more traditional approaches of using implicit-explicit (IMEX) and low-order operator splitting schemes. However, to date there have been few performance studies that compare different classes of multirate integrators on complex application problems. In this work, we study the performance of several newly developed multirate infinitesimal (MRI) methods, implemented in the SUNDIALS solver package, on two reacting flow model problems built on structured mesh frameworks. The first model revisits prior work on a compressible reacting flow problem with complex chemistry that is implemented using BoxLib but where we now include comparisons between a new explicit MRI scheme with the multirate spectral deferred correction (SDC) methods in the original paper. The second problem uses the same complex chemistry as the first problem, combined with a simplified flow model, but runs at a large spatial scale where explicit methods become infeasible due to stability constraints. Two recently developed IMEX MRI multirate methods are tested. These methods rely on advanced features of the AMReX framework on which the model is built, such as multilevel grids and multilevel preconditioners. The results from these two problems show that MRI multirate methods can offer significant performance benefits on complex multiphysics application problems and that these methods may be combined with advanced spatial discretization to compound the advantages of both.

Implicit–explicit multirate infinitesimal stage-restart methods

Implicit–Explicit (IMEX) methods are flexible numerical time integration methods which solve an initial-value problem (IVP) that is split into stiff and nonstiff processes with the goal of lower computational costs than a purely implicit or explicit approach. A complementary form of flexible IVP solvers are multirate infinitesimal methods for problems split into fast- and slow-changing dynamics, that solve a multirate IVP by evolving a sequence of “fast” IVPs using any suitably accurate algorithm. This article introduces a new class of high-order implicit–explicit multirate methods that are designed for multirate IVPs in which the slow-changing dynamics are further split in an IMEX fashion. This new class, which we call implicit–explicit multirate infinitesimal stage-restart (IMEX-MRI-SR), both improves upon the previous implicit–explicit multirate infinitesimal generalized-structure additive Runge Kutta (IMEX-MRI-GARK) methods by allowing for far easier creation of new embedded methods, and extends multirate exponential Runge Kutta (MERK) methods by allowing the fast-changing dynamics to be nonlinear and the methods to be implicit. We leverage GARK theory to derive conditions for orders of accuracy up to four, and we provide second- and third-order accurate example methods, which are the first known embedded MRI methods with IMEX structure. We then perform numerical simulations demonstrating convergence rates and computational performance in both fixed-step and adaptive-step settings.

A time adaptive multirate Dirichlet–Neumann waveform relaxation method for heterogeneous coupled heat equations

AbstractWe consider partitioned time integration for heterogeneous coupled heat equations. First and second order multirate, as well as time‐adaptive Dirichlet‐Neumann Waveform relaxation (DNWR) methods are derived. In 1D and for implicit Euler time integration, we analytically determine optimal relaxation parameters for the fully discrete scheme.We test the robustness of the relaxation parameters on the second order multirate method in 2D. DNWR is shown to be very robust and consistently yielding fast convergence rates, whereas the closely related Neumann‐Neumann Waveform relaxtion (NNWR) method is slower or even diverges.The waveform approach naturally allows for different timesteps in the subproblems. In a performance comparison for DNWR, the time‐adaptive method dominates the multirate method due to automatically finding suitable stepsize ratios. Overall, we obtain a fast, robust, multirate and time adaptive partitioned solver for unsteady conjugate heat transfer.

Multirate Exponential Rosenbrock Methods

In this paper we propose a novel class of methods for high order accurate integration of multirate systems of ordinary differential equation initial-value problems. The proposed methods construct multirate schemes by approximating the action of matrix $\varphi$-functions within explicit exponential Rosenbrock (ExpRB) methods, thereby called Multirate Exponential Rosenbrock (MERB) methods. They consist of the solution to a sequence of modified "fast" initial-value problems, that may themselves be approximated through subcycling any desired IVP solver. In addition to proving how to construct MERB methods from certain classes of ExpRB methods, we provide rigorous convergence analysis of these methods and derive efficient MERB schemes of orders two through six (the highest order ever constructed infinitesimal multirate methods). We then present numerical simulations to confirm these theoretical convergence rates, and to compare the efficiency of MERB methods against other recently-introduced high order multirate methods.

Entropy–Preserving and Entropy–Stable Relaxation IMEX and Multirate Time–Stepping Methods

We propose entropy-preserving and entropy-stable partitioned Runge–Kutta (RK) methods. In particular, we extend the explicit relaxation Runge–Kutta methods to IMEX–RK methods and a class of explicit second-order multirate methods for stiff problems arising from scale-separable or grid-induced stiffness in a system. The proposed approaches not only mitigate system stiffness but also fully support entropy-preserving and entropy-stability properties at a discrete level. The key idea of the relaxation approach is to adjust the step completion with a relaxation parameter so that the time-adjusted solution satisfies the entropy condition at a discrete level. The relaxation parameter is computed by solving a scalar nonlinear equation at each timestep in general; however, as for a quadratic entropy function, we theoretically derive the explicit form of the relaxation parameter and numerically confirm that the relaxation parameter works the Burgers equation. Several numerical results for ordinary differential equations and the Burgers equation are presented to demonstrate the entropy-conserving/stable behavior of these methods. We also compare the relaxation approach and the incremental direction technique for the Burgers equation with and without a limiter in the presence of shocks.

Multirate DAE-simulation and its application in system simulation software for the development of electric vehicles

This work is devoted to the efficient simulation of large multi-physical networks stemming from automated modelling processes in system simulation software. The simulation of hybrid, battery and fuel cell electric vehicle applications requires the coupling of electrical, mechanical, fluid, and thermal networks. Each network is established by combining the connection structure of a graph with physical equations of elementary components and resulting in a DAE (differential algebraic equation). To speed up the simulation a non-iterative multirate time integration co-simulation method for the system of coupled DAEs is established. The power of the multirate method is shown via two representative examples of battery powered electric vehicle with a cooling system for the battery pack and a three phase inverter with a cooling system. This work is an extended version of (Kolmbauer et al. in Scientific Computing in Electrical Engineering (SCEE 2020), Springer, Cham, pp. 231–240, 2021).

Explicit stabilized multirate method for stiff differential equations

Stabilized Runge–Kutta methods are especially efficient for the numerical solution of large systems of stiff nonlinear differential equations because they are fully explicit. For semi-discrete parabolic problems, for instance, stabilized Runge–Kutta methods overcome the stringent stability condition of standard methods without sacrificing explicitness. However, when stiffness is only induced by a few components, as in the presence of spatially local mesh refinement, their efficiency deteriorates. To remove the crippling effect of a few severely stiff components on the entire system of differential equations, we derive a modified equation, whose stiffness solely depends on the remaining mildly stiff components. By applying stabilized Runge–Kutta methods to this modified equation, we then devise an explicit multirate Runge–Kutta–Chebyshev (mRKC) method whose stability conditions are independent of a few severely stiff components. Stability of the mRKC method is proved for a model problem, whereas its efficiency and usefulness are demonstrated through a series of numerical experiments.

Explicit Stabilized Multirate Method for Stiff Stochastic Differential Equations

Stabilized explicit methods are particularly efficient for large systems of stiff stochastic differential equations (SDEs) due to their extended stability domain. However, they loose their efficiency when a severe stiffness is induced by very few fast degrees of freedom, as the stiff and nonstiff terms are evaluated concurrently. Therefore, inspired by [A. Abdulle, M. J. Grote, and G. Rosilho de Souza, Preprint (2020), arXiv:2006.00744] we introduce a stochastic modified equation whose stiffness depends solely on the slow terms. By integrating this modified equation with a stabilized explicit scheme we devise a multirate method which overcomes the bottleneck caused by a few severely stiff terms and recovers the efficiency of stabilized schemes for large systems of nonlinear SDEs. The scheme is not based on any scale separation assumption of the SDE and therefore it is employable for problems stemming from the spatial discretization of stochastic parabolic partial differential equations on locally refined grids. The multirate scheme has strong order 1/2, weak order 1 and its stability is proved on a model problem. Numerical experiments confirm the efficiency and accuracy of the scheme.

Multirate Partially Explicit Scheme for Multiscale Flow Problems

For time-dependent problems with high-contrast multiscale coefficients, the time step size for explicit methods is affected by the magnitude of the coefficient parameter. With a suitable construction of multiscale space, one can achieve a stable temporal splitting scheme where the time step size is independent of the contrast. Consider the parabolic equation with heterogeneous diffusion parameter, the flow rates vary significantly in different regions due to the high-contrast features of the diffusivity. In this work, we aim to introduce a multirate partially explicit splitting scheme to achieve efficient simulation with the desired accuracy. We first design multiscale subspaces to handle flow with different speed. For the fast flow, we obtain a low-dimensional subspace with respect to the high-diffusive component and adopt an implicit time discretization scheme. The other multiscale subspace will take care of the slow flow, and the corresponding degrees of freedom are treated explicitly. Then a multirate time stepping is introduced for the two parts. The stability of the multirate methods is analyzed for the partially explicit scheme. Moreover, we derive local error estimators corresponding to the two components of the solutions and provide an upper bound of the errors. An adaptive local temporal refinement framework is then proposed to achieve higher computational efficiency. Several numerical tests are presented to demonstrate the performance of the proposed method.

Asynchronous multi‐rate method of real‐time simulation for active distribution networks

The real-time simulation of active distribution networks (ADNs) can provide an accurate insight into transient behaviours, but faces challenges in simulation efficiency and flexibility brought by larger system scales and wider time-scale ranges. This paper presents an asynchronous multi-rate (AMR) method and design for the real-time simulation of large-scale ADNs. In the proposed method, the entire ADN was decoupled into different subsystems according to accuracy requirements, and optimized time-steps were allocated to each subsystem to realize a fully distributed simulation. This not only alleviated the time-step coordination problem existing in multi-rate real-time simulations, but also enhanced the flexible expansion capabilities of the real-time simulator. To realize the AMR real-time simulation, a multi-rate interfacing method, synchronization mechanism, and data communication strategy are proposed in this paper, and their hardware design is also presented in detail. A modified IEEE 123-node system with photovoltaics and wind turbine generators was simulated on a 3 field-programmable gate arrays (FPGAs)-based AMR real-time simulator. The real-time results were captured by the oscilloscope and verified with PSCAD/EMTDC, which demonstrated the superiority in simulation flexibility and accuracy compared with the synchronous multi-rate (SMR) method.

Implicit extensions of an explicit multirate Runge–Kutta scheme

We propose a new method that extends conservative explicit multirate methods to implicit explicit-multirate methods. We develop extensions of order one and two with different stability properties on the implicit side. The method is suitable for time-stepping adaptive mesh refinement PDE discretizations with different degrees of stiffness. A numerical example with an advection–diffusion problem illustrates the new method’s properties.

Multirate linearly-implicit GARK schemes

Many complex applications require the solution of initial-value problems where some components change fast, while others vary slowly. Multirate schemes apply different step sizes to resolve different components of the system, according to their dynamics, in order to achieve increased computational efficiency. The stiff components of the system, fast or slow, are best discretized with implicit base methods in order to ensure numerical stability. To this end, linearly implicit methods are particularly attractive as they solve only linear systems of equations at each step. This paper develops the Multirate GARK-ROS/ROW (MR-GARK-ROS/ROW) framework for linearly-implicit multirate time integration. The order conditions theory considers both exact and approximative Jacobians. The effectiveness of implicit multirate methods depends on the coupling between the slow and fast computations; an array of efficient coupling strategies and the resulting numerical schemes are analyzed. Multirate infinitesimal step linearly-implicit methods, that allow arbitrarily small micro-steps and offer extreme computational flexibility, are constructed. The new unifying framework includes existing multirate Rosenbrock(-W) methods as particular cases, and opens the possibility to develop new classes of highly effective linearly implicit multirate integrators.

Block-based compressed sensing of MR images using multi-rate deep learning approach

Deep learning (DL) models are highly research-oriented field in image compressive sensing in the recent studies. In compressive sensing theory, a signal is efficiently reconstructed from very small and limited number of measurements. Block-based compressive sensing is most promising and lenient compressive sensing (CS) approach mostly used to process large-sized videos and images: exploit low computational complexity and requires less memory. In block-based compressive sensing, a number of deep models are needed to train with each corresponding to different sampling rate. Compressive sensing performance is highly degraded through allocating low sampling rates to various blocks within same image or video frames. In this work, we proposed multi-rate method using deep neural networks for block-based compressive sensing of magnetic resonance images with performance that greatly outperforms existing state-of-the-art methods. The proposed approach is capable in smart allocation of exclusive sampling rate for each block within image, based on the image information and removing blocking artifacts in reconstructed MRI images. Each image block is separately sampled and reconstructed with different sampling rate and reassembled into a single image based on inter-correlation between blocks, to remove blocking artifacts. The proposed method surpasses the current state-of-the-arts in terms of reconstruction speed, reconstruction error, low computational complexity, and certain evaluation metrics such as peak signal-to-noise ratio (PSNR), structural similarity (SSIM), feature similarity (FSIM), and relative l2-norm error (RLNE).

Implicit Multirate GARK Methods

This work considers multirate generalized-structure additively partitioned Runge–Kutta methods for solving stiff systems of ordinary differential equations with multiple time scales. These methods treat different partitions of the system with different timesteps for a more targeted and efficient solution compared to monolithic single rate approaches. With implicit methods used across all partitions, methods must find a balance between stability and the cost of solving nonlinear equations for the stages. In order to characterize this important trade-off, we explore multirate coupling strategies, problems for assessing linear stability, and techniques to efficiently implement Newton iterations for stage equations. Unlike much of the existing multirate stability analysis which is limited in scope to particular methods, we present general statements on stability and describe fundamental limitations for certain types of multirate schemes. New implicit multirate methods up to fourth order are derived, and their accuracy and efficiency properties are verified with numerical tests.

Implicit-Explicit Multirate Infinitesimal GARK Methods

This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. Unlike other recent work in this area, the proposed methods support mixed implicit-explicit (IMEX) treatment of the slow time scale. In addition to allowing this slow time scale flexibility, the proposed methods utilize a so-called `infinitesimal' formulation for the fast time scale through definition of a sequence of modified `fast' initial-value problems, that may be solved using any viable algorithm. We name the proposed class as implicit-explicit multirate infinitesimal generalized-structure additive Runge--Kutta (IMEX-MRI-GARK) methods. In addition to defining these methods, we prove that they may be viewed as specific instances of GARK methods, and derive a set of order conditions on the IMEX-MRI-GARK coefficients to guarantee both third and fourth order accuracy for the overall multirate method. Additionally, we provide three specific IMEX-MRI-GARK methods, two of order three and one of order four. We conclude with numerical simulations on two multirate test problems, demonstrating the methods' predicted convergence rates and comparing their efficiency against both legacy IMEX multirate schemes and recent third and fourth order implicit MRI-GARK methods.

Thermal Behavior of the Low-temperature Secondary Oxidation of Coal under Different Pre-oxidation Temperatures

ABSTRACT To study the effect of pre-oxidation temperature on the thermal behavior of secondary oxidation of coal, this paper studies samples from a Jurassic coalfield that is susceptible to spontaneous combustion. Under an air atmosphere, pre-oxidized coal samples with different pre-oxidation temperatures (75°C, 115°C, 155°C, 195°C) were prepared by heating at a rate of 0.5°C·min−1. Samples of each, and a raw coal sample, were then heated from 30–300°C at heating rates of 0.2, 0.3, 0.4, and 0.5°C·min−1 in an air atmosphere of 120 mL·min−1. The micro-characteristics of pre-oxidized coal samples were identified by Fourier infrared spectroscopy method. A C80 microcalorimeter was employed to monitor the thermal behavior of low-temperature secondary oxidation of coal. The heat flow curves of raw coal and pre-oxidized coal samples during the heating process show similar trends. Characteristic temperatures determined from the data of samples from different endothermic stages increase with increasing pre-oxidation temperature. The heat absorption of pre-oxidized coal samples in different endothermic stages is significantly lower than that of raw coal, and decreases first and then increases with increasing pre-oxidation temperature. Due to the effect of pre-oxidation, the methyl, hydroxyl and carboxyl groups of pre-oxidized coal samples are obviously lower than that of raw coal. The apparent activation energy of different oxidation stages of the experimental coal samples was calculated with a multi-rate heating method. The secondary oxidation activation energy of coal increases first and then decreases with increasing pre-oxidation temperature. In the rapid endothermic stage and the slow endothermic stage, the activation energy ranges of the coal samples were 68.12–201.46 kJ·mol−1 and 26.21–135.58 kJ·mol−1.

Multi-Rate Real-Time Simulation Method Based on the Norton Equivalent

For the problem of poor accuracy of the existing multi-rate simulation methods, this paper proposes a multi-rate real-time simulation method based on the Norton equivalent, compared with multi-rate simulation method based on the ideal source equivalent. After the Norton equivalence of the fast subsystem and the slow subsystem are established, they are solved simultaneously at the junction nodes. In order to reduce the amount of the simulation calculation, the Norton equivalent circuit is obtained by incremental calculation. The data interaction between the fast subsystem and the slow subsystem is realized by extrapolation method. For ensuring the real-time performance of the simulation, the method of the slow subsystem calculates ahead of the fast subsystem is given for the slow subsystem with a large amount of calculation. Finally, the AC/DC hybrid power system was simulated on the real-time simulation platform (FPGA-based Real-Time Digital Solver, FRTDS), and the simulation results were compared with the single-rate simulation, which verified the correctness and accuracy of the proposed method.

Predicting coupling signals in a material flow real-time co-simulation with a Kalman filter

A hardware-in-the-loop simulation has to use the control cycle as simulation clock. This time-deterministic clocking can be difficult to meet for complex behavior models in a real-time simulation. To accelerate a real-time simulation, the calculation can be parallelized by co-simulation whereby multiple, differently detailed behavior models can be simulated parallelly. In these multi-scale models, a slower, detailed model guides a fast, abstract model. In this work, a Kalman filter is examined as a multi-rate method to predict the signals of a physical-based model for a material flow simulation in real-time. To improve the results, smoothing and boundary conditions are added.

A self adjusting multirate algorithm for robust time discretization of partial differential equations

We show the distinctive potential advantages of a self adjusting multirate method based on diagonally implicit solvers for the robust time discretization of partial differential equations. The properties of the specific ODE methods considered are reviewed, with special focus on the TR-BDF2 solver. A general expression for the stability function of a generic one stage multirate method is derived, which allows to study numerically the stability properties of the proposed algorithm in a number of examples relevant for applications. Several numerical experiments, aimed at the time discretization of hyperbolic partial differential equations, demonstrate the efficiency and accuracy of the resulting approach.